This was probably true for most of human history but I specifically referenced successful civilizations. The way that humans lived before creating civilization isn't really germane to the topic at hand. Well, how? It most certainly hasn't been. We HAD a system that helped to minimize single-parent situations and it worked for literally thousands of years. When we dropped it the upper classes still continued by-and-large to have kids as married couples but the HS dropouts:
Big issue I've got here is that to an extent I think you're creating a concept of a "we" that "acts".
"We" didn't have a system. Religion existed, a lot of people in America believed in it, and so more people conformed to religious norms than today. But "we" didn't drop it. Religion still existed, fewer people in America believed in it, and so fewer people conformed to religious norms than before.
Nobody decided to change the system. People individually changed. And now you're looking at aggregate statistics of the results of that change as if the "system" changed.
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Second thing I'd say is that I highly doubt we can point to the rise in unmarried mothers as being due to a lack of religion. You do point out some increases in single mother families across demographics, but those increases MASSIVELY dwarf any rise in atheism.
This is particularly egregious from a statistical standpoint because religious belief seems to be more sticky in the lower classes:
https://www.pewresearch.org/religion/religious-landscape-study/income-distribution/For people with household incomes <$50K, 85% believe with absolute or reasonable certainty that god exists. That number drops to 75% with the >$100K crowd, and the number that affirmatively state they don't believe has jumped from 7-9% for those <$50K up to 14% >$100K.
IMHO a growth in the lack of religious belief may be *a* factor in what is going on, but it doesn't seem, demographically, likely to be the *dominant* factor.